ENH: Implement squared exponential covariance function

Implement squared exponential covariance function.

src/nifreeze/model/gpr.py

@@ -30,6 +30,7 @@
 import numpy as np
 from scipy import optimize
 from scipy.optimize._minimize import Bounds
+from scipy.spatial.distance import cdist, pdist, squareform
 from sklearn.gaussian_process import GaussianProcessRegressor
 from sklearn.gaussian_process.kernels import (
     Hyperparameter,
@@ -40,6 +41,8 @@
 
 BOUNDS_A: tuple[float, float] = (0.1, 2.35)
 """The limits for the parameter *a* (angular distance in rad)."""
+BOUNDS_ELL: tuple[float, float] = (0.1, 2.35)
+"""The limits for the parameter *$\ell$* (shell distance in $s/mm^2$)."""
 BOUNDS_LAMBDA: tuple[float, float] = (1e-3, 1000)
 """The limits for the parameter λ (signal scaling factor)."""
 THETA_EPSILON: float = 1e-5
@@ -469,6 +472,121 @@ def __repr__(self) -> str:
         return f"SphericalKriging (a={self.beta_a}, λ={self.beta_l})"
 
 
+class SquaredExponentialKriging(Kernel):
+    """A scikit-learn's kernel for DWI signals."""
+
+    def __init__(
+        self,
+        beta_ell: float = 1.38,
+        beta_l: float = 0.5,
+        ell_bounds: tuple[float, float] = BOUNDS_ELL,
+        l_bounds: tuple[float, float] = BOUNDS_LAMBDA,
+    ):
+        r"""
+        Initialize a spherical Kriging kernel.
+
+        Parameters
+        ----------
+        beta_ell : :obj:`float`, optional
+            Minimum angle in rads.
+        beta_l : :obj:`float`, optional
+            The :math:`\lambda` hyperparameter.
+        ell_bounds : :obj:`tuple`, optional
+            Bounds for the :math:`\ell`  parameter.
+        l_bounds : :obj:`tuple`, optional
+            Bounds for the :math:`\lambda` hyperparameter.
+
+        """
+        self.beta_ell = beta_ell
+        self.beta_l = beta_l
+        self.a_bounds = ell_bounds
+        self.l_bounds = l_bounds
+
+    @property
+    def hyperparameter_ell(self) -> Hyperparameter:
+        return Hyperparameter("beta_ell", "numeric", self.a_bounds)
+
+    @property
+    def hyperparameter_l(self) -> Hyperparameter:
+        return Hyperparameter("beta_l", "numeric", self.l_bounds)
+
+    def __call__(
+        self, X: np.ndarray, Y: np.ndarray | None = None, eval_gradient: bool = False
+    ) -> np.ndarray | tuple[np.ndarray, np.ndarray]:
+        """
+        Return the kernel K(X, Y) and optionally its gradient.
+
+        Parameters
+        ----------
+        X : :obj:`~numpy.ndarray`
+            Gradient wighting values (X)
+        Y : :obj:`~numpy.ndarray`, optional
+            Gradient wighting values (Y, optional)
+        eval_gradient : :obj:`bool`, optional
+            Determines whether the gradient with respect to the log of
+            the kernel hyperparameter is computed.
+            Only supported when Y is ``None``.
+
+        Returns
+        -------
+        K : :obj:`~numpy.ndarray` of shape (n_samples_X, n_samples_Y)
+            Kernel k(X, Y)
+
+        K_gradient : :obj:`~numpy.ndarray` of shape (n_samples_X, n_samples_X, n_dims),\
+                optional
+            The gradient of the kernel k(X, X) with respect to the log of the
+            hyperparameter of the kernel. Only returned when ``eval_gradient``
+            is True.
+
+        """
+
+        dists = compute_shell_distance(X, Y=Y)
+        C_b = squared_exponential_covariance(dists, self.beta_ell)
+
+        if Y is None:
+            C_b = squareform(C_b)
+            np.fill_diagonal(C_b, 1)
+
+        if not eval_gradient:
+            return self.beta_l * C_b
+
+        # Looking at this
+        # https://github.com/scikit-learn/scikit-learn/blob/1e6a81f322f1821cc605a18b08fcc198c7d63c97/sklearn/gaussian_process/kernels.py#L1574
+        # Not sure the derivative is clear to me. IMO it should be
+        # \frac{d}{dx} \left( e^{-\frac{cte}{x^2}} \right) = \frac{2 cte}{x^3} \cdot e^{-\frac{cte}{x^2}}
+        # where x is ell, and cte is 0.5 * (\log b - \log b')^2
+
+        K_gradient = 1 # ToDo
+
+        return self.beta_l * C_b, K_gradient
+
+    def diag(self, X: np.ndarray) -> np.ndarray:
+        """Returns the diagonal of the kernel k(X, X).
+
+        The result of this method is identical to np.diag(self(X)); however,
+        it can be evaluated more efficiently since only the diagonal is
+        evaluated.
+
+        Parameters
+        ----------
+        X : :obj:`~numpy.ndarray` of shape (n_samples_X, n_features)
+            Left argument of the returned kernel k(X, Y)
+
+        Returns
+        -------
+        K_diag : :obj:`~numpy.ndarray` of shape (n_samples_X,)
+            Diagonal of kernel k(X, X)
+        """
+        return self.beta_l * np.ones(X.shape[0])
+
+    def is_stationary(self) -> bool:
+        """Returns whether the kernel is stationary."""
+        return True
+
+    def __repr__(self) -> str:
+        return f"SquaredExponentialKriging (wll={self.beta_ell}, λ={self.beta_l})"
+
+
 def exponential_covariance(theta: np.ndarray, a: float) -> np.ndarray:
     r"""
     Compute the exponential covariance for given distances and scale parameter.
@@ -590,3 +708,71 @@ def compute_pairwise_angles(
     thetas = np.arccos(np.abs(cosines)) if closest_polarity else np.arccos(cosines)
     thetas[np.abs(thetas) < THETA_EPSILON] = 0.0
     return thetas
+
+
+def squared_exponential_covariance(
+    shell_distance: np.ndarray,
+    ell: float,
+) -> np.ndarray:
+    r"""Compute the squared exponential covariance for given diffusion gradient
+    encoding weighting distances and scale parameter.
+
+    Implements :math:`C_{b}`, following Eq. (15) in [Andersson15]_:
+
+    .. math::
+
+        C_{b}(b, b'; \ell) = \exp\left( - \frac{(\log b - \log b')^2}{2 \ell^2} \right)
+
+    The squared exponential covariance function is sometimes called radial basis
+    function (RBF) or Gaussian kernel.
+
+    Parameters
+    ----------
+    shell_distance : :obj:`~numpy.ndarray` of shape (n_samples_X, n_features)
+        Input data.
+    ell : float
+        Distance parameter where the covariance function goes to zero.
+
+    Returns
+    -------
+    :obj:`~numpy.ndarray`
+        Squared exponential covariance values for the input distances.
+    """
+
+    return np.exp(-0.5 * (shell_distance / (ell**2)))
+
+
+def compute_shell_distance(X, Y=None):
+    r"""Compute pairwise angles across diffusion gradient encoding weighting
+    values.
+
+   Following Eq. (15) in [Andersson15]_, computes the distance between the log
+   values of the diffusion gradient encoding weighting values:
+
+    .. math::
+
+        \log b - \log b'
+
+    Parameters
+    ----------
+    X : :obj:`~numpy.ndarray` of shape (n_samples_X, n_features)
+        Input data.
+    Y : :obj:`~numpy.ndarray` of shape (n_samples_Y, n_features), optional
+        Input data. If ``None``, the output will be the pairwise
+        similarities between all samples in ``X``.
+
+    Returns
+    -------
+    :obj:`~numpy.ndarray`
+        Pairwise distances of diffusion gradient encoding weighting values.
+    """
+
+    # ToDo
+    # scikit-learn RBF call includes $\ell$ here; fine, but then I do not get
+    # the derivative computation the way they compute it
+    if Y is None:
+        dists = pdist(np.log(X), metric="sqeuclidean")
+    else:
+        dists = cdist(np.log(X), np.log(Y), metric="sqeuclidean")
+
+    return dists

test/test_gpr.py

@@ -271,8 +271,8 @@ def test_compute_pairwise_angles(bvecs1, bvecs2, closest_polarity, expected):
     np.testing.assert_array_almost_equal(obtained, expected, decimal=2)
 
 
-@pytest.mark.parametrize("covariance", ["Spherical", "Exponential"])
-def test_kernel(repodata, covariance):
+@pytest.mark.parametrize("covariance", ["Spherical", "Exponential", "SquaredExponential"])
+def test_kernel_single_shell(repodata, covariance):
     """Check kernel construction."""
 
     bvals, bvecs = read_bvals_bvecs(
@@ -296,3 +296,65 @@ def test_kernel(repodata, covariance):
 
     K_predict = kernel(bvecs, bvecs[10:14, ...])
     assert K_predict.shape == (K.shape[0], 4)
+
+
+# ToDo
+@pytest.mark.parametrize(
+    ("bvals1", "bvals2", "expected"),
+    [
+        (
+            np.array(
+                [
+                    [1000, 1000, 1000, 1000],
+                ]
+            ),
+            None,
+            np.array(
+                [
+                    [0, 0, 0, 0],
+                ]
+            ),
+        ),
+        (
+            np.array(
+                [
+                    [1000, 1000, 1000, 1000],
+                    [2000, 2000, 2000],
+                ]
+            ),
+            None,
+            np.array(
+                [
+                    [1000, 1000, 1000, 1000],
+                ]
+            ),
+        ),
+    ],
+)
+def test_compute_shell_distance(bvals1, bvals2, expected):
+
+    obtained = gpr.compute_shell_distance(bvals1, bvals2)
+
+    if bvals2 is not None:
+        assert (bvals1.shape[0], bvals2.shape[0]) == obtained.shape
+    assert obtained.shape == expected.shape
+    np.testing.assert_array_almost_equal(obtained, expected, decimal=2)
+
+
+@pytest.mark.parametrize("covariance", ["SquaredExponential"])
+def test_kernel_multi_shell(repodata, covariance):
+    """Check kernel construction."""
+
+    bvals, bvecs = read_bvals_bvecs(
+        str(repodata / "ds000114_multishell.bval"),
+        str(repodata / "ds000114_multishell.bvec"),
+    )
+
+    bvals = bvals[bvals > 10]
+
+    KernelType = getattr(gpr, f"{covariance}Kriging")
+    kernel = KernelType()
+
+    K = kernel(bvals)
+
+    assert K.shape == (bvals.shape[0],) * 2